A wiki is a website that allows for rapid creation and editing of pages (often called articles in a wiki context) through a web browser, instead of requiring direct access to the server hosting the wiki. Most wikis further attempt to speed up the editing process by defining a simple markup syntax to be used for formatting which is (ideally) faster and easier to use than HTML. Wiki pages are often loaded dynamically using a server side scripting language like PHP or ASP in order to convert the stored wiki-markup into the appropriate HTML for viewing.
Wikis are conduscive to situations in which multiple persons need to edit the website, potentially from very remote locations. All one needs to edit the wiki (assuming they have permission to do so) is a web browser which can connect to the wiki site. The site itself has built in functionality for editing the content. Furthermore, many wikis record a history of changes made to each page, allowing old versions of a page to be referenced and restored; essentially built-in version control.
Wiki's are basically just a collection of records (about pages, users, histories, settings, etc.), and a collection of scripts to keep them organized and provide the web interface for viewing, editing, and managing the site. The package of scripts and other software or files that do this is the wiki engine, or wiki software.
The site you're looking at right now is a wiki. In this case, the wiki is being hosted on my home webserver, using the MediaWiki engine.
